Relaxation of Hot Quasiparticles in a d-WaveSuperconductor
A. Rosch (Cologne), P. Howell (Karlsruhe)
P. Hirschfeld (Florida)
• pump-probe experiments in high-temperature superconductors
• relaxation of high-energy quasi-particles
• momentum conservation and diffusion in momentum space
Phase Diagram of the Cuprates
Fermi liquid?
T
AF
M in
sulato
r
pseudogap
strange metal
hole concentration
superconductor
best understood (??): physics within superconducting phase
Scenario 1: Quasiparticles in superconducting phase
Microwave conductivity σ(ω) in
YBa2Cu3O6−δ Hosseini et al. (1999)
well fitted by thermal QPs
Agreement with λ−2(T).
τ−1tr decreases fast below Tc
Umklapp scattering of QPs
Walker & Smith, PRB 61, 11285 (2000)
Scenario 2: Quantum Critical Scaling
σ(ω) in Bi2Sr2CaCu2O8 requires more complicated model
→ ~τ−1qp ≃ 0.8kBT
This agrees with ARPES:
Nodal QPs of energy ω show ‘quan-
tum critical scaling’:
~τ−1qp ≃ max[~ω, kBT ]
above and below Tc
(marginal Fermi liquid ??)
♦ = 48 K, △ = 90 K, ◦ = 300 KValla et al., Science 285, 2110 (1999)
Pump–Probe Experiments in Superconductors
Pump with laser pulse — probe optical response
cascade of pair−breaking down to gap
phonons equilibrated with hot QP transport energy out of system
Step 0: excitation of QP (1 eV)
Step 1 ( < 1ps):
Step 2 (1−10ps):
Step 3 (ns − s):
This talk !
µ
Rothwarf, Taylor (1967)
Pump–Probe Reflectivity on YBa2Cu3O6.5
time-resolved reflectivity δR ∝ ∫
dω ω2δσ(ω) after pumping
Segre, Gedik, Orenstein, Bonn, Liang, Hardy et al., PRL 88, 137001 (2002)
QPs relax very fast down ∼∆0, then slowly with τ−1 ∝ T3
Mystery:QP energy ∼∆0 ≫ T so expect τ independent of T.
More experiments...
in general: very different time-scales measured in pump-probe experi-ments in HTc depending on doping, sample quality, frequency, inten-sity, temperature
some recent experiments:
• Diffusion of nonequilibrium quasi-particles in a cuprate superconductor, (Gedik et
al. Science 2003):writing a interference pattern on crystal to obtain diffusion
• Dynamics of Cooper pair formation in Bi2Sr2CaCu2O8−δ (Kaindl et al.,Science 2003,
PRB, 2005):time-dependence of σ(4meV < ω < 11meV )
• Nonequilibrium quasiparticle relaxation in the vortex state
of La2−xSrxCuO4, (Bianchi et al. PRL 2005)
• Abrupt transition in quasiparticle dynamics at op-
timal doping in a cuprate superconductor system
(Gedik, et al. PRL 2005): different sign and differ-ent density dependence for under- and overdopedsamples!
underdoping vs. overdoping (Gedik, et al. PRL 2005)
need other hot QP to free trapped excitations for underdoped system?
Fermi Liquid Interpretation
Photoinduced QPs create cascade of QPs, i.e. excite pairs out of
condensate.
Some QPs end up near gap antinodes with energy ∆0.
How do these ‘hot’ QPs relax?
E and k conservation ⇒bottleneck for pair-breaking
processes.
Intermediate T:
Umklapp scattering from thermal
QPs with activation gap . ∆0.
Low T:
Slow relaxation by scattering from
thermal QPs at nodes.
antinode
thermal QPs
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NB: Clean crystals ⇒ negligible impurity scattering
Bottleneck for Pair-Breaking
Scattering process d†−p−qd
†pd
†k+qdk
Idea: antinodal velocity < nodal velocity ⇒antinodal QP cannot lose enough energy to generate nodal QP-pairs
EANk = ∆0 cos 2θ +
2k2E2F
∆0 cos 2θ
ENp = 2
√
E2Fp2
⊥ + ∆20p
2‖
Energy lost by antinodal QP:
EANk − EAN
k+q = 2q∆0 sin 2θ
Energy needed to create pair:
/4
p
p
q
θα
−p−q
k
ENp + EN
−p−q
= 2
√
E2Fq2 cos2 α + ∆2
0q2 sin2 α ≃ 2qEF cos(π
4 + θ)
⇒ Pair-breaking only possible if θ ≃ π4
No scattering for |k − kF| .1
√2
∆20
E2F
cos2 2θ
How else can Quasiparticles Relax?
Also bottleneck for phonons and spin waves.
What about other QP scattering processes?
Hint =∑
1,2,3,4
[
V2−3(u1v2 + v1u2)(u3v4 + v3u4)d†4↑d
†3↓d
†2↑d
†1↓
+ V2+1(u1u2 − v1v2)(u3v4 + v3u4)d†4↑d
†3↓d
†2σd1σ
+ V2+1(u1v2 + v1u2)(u3v4 + v3u4)d†4σd
†3σ̄d2σ̄d1σ
+ V2−3(u1u4 − v1v4)(u2u3 − v2v3)d†4σd
†3σ′d2σ′d1σ
+ h.c.]
Consider QP–QP scattering terms:
Low density of hot QPs, so scatter off thermal QPs.
Umklapp Scattering
QP–QP scattering: d†k+qd
†p−q+Gdpdk
Large momentum transfer possible, q ∼ kF, but thermal QP can’t bearbitrarily close to node
⇒ τ−1U ∼ e−Eact/T with activation energy Eact . ∆0
(−π,−π)
(−π,π)
(π,−π)
(π,π)
"hot" QP
QPthermal
gap
G=(2 ,2 )
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π π exponentially suppressed
at low T!
Quasi-Elastic Scattering
Scattering from nodal QPs: d†k+qd
†p−qdpdk
Discrepancy between nodal and
antinodal velocities
⇒ q = (q‖,q⊥) ∼ ( T∆0
, TEF
) ≪ kF
p
p
/4π
k
p
q
−q
small transferred momentum ⇒ diffusion in momentum space
hot QP reaches quasi-equilibrium with nodal QPs: |k−kF | ∼ √T∆0/EF
• for T ≪ ∆30
E2F
⇒ |k − kF| ≪∆2
0E2
F
⇒ hot QP inside bottleneck
• for T ≫ ∆30
E2F
⇒ |k − kF| ≫∆2
0E2
F
⇒ hot QP outside bottleneck
Boltzmann Equation at T ≪ ∆30/E2
F (1)
Calculate evolution of antinodal distribution gk(t).
∂gk
∂t= −
∫
dq[
gkGqq(q, Ek − Ek+q) − gk−qGqq(q, Ek−q − Ek)]
,
Gqq(q, ε) = |Vqq|2∫
dp fp(1 − fp−q) δ(Ep − Ep−q + ε)
Hot QPs ‘thermalise’ in radial direction → transformation:
gk(t) =Ck(t)
hθe−βEk , hθ =
∫
(kF + k)dk e−βEk
Quasi-equilibrium in radial direction
⇒ Ck only weakly dependent on |k|.
Ck = C0θ + (k − 〈k〉)C1
θ + · · ·
Boltzmann Equation at T ≪ ∆30/E2
F (2)
Project out C0θ and do gradient expansion since q ≪ kth.
∂
∂t
[
C0θ(t)
hθ
]
=
[
Fθ∂
∂θ+ D
∂2
∂θ2
]
C0θ(t)
hθ(1)
where F = 4β∆0D =7π5
80
|Vqq|2T4
∆30E
2F
Recall gk(t) =Ck(t)
hθe−βEk Equilibrium? C0
θ(t) = hθ.√
Particle conservation? N =∫
dkgk =∫
dθ C0θ
Re-write (1) as
∂
∂tC0
θ = −F∂
∂θ
(
θC0θ
)
+ D∂2
∂θ2C0
θ
Right-hand side has only gradient terms.√
⇒ Structure of equation + ratio F/D fixed by symmetries!
Diffusion Equation
∂
∂tC0
θ = −F∂
∂θ
(
θC0θ
)
︸ ︷︷ ︸
force due to energygradient from antinode
+D∂2
∂θ2C0
θ︸ ︷︷ ︸
diffusion
Dominant contribution from φ ≃ π4
⇒ Zig-zig diffusion in momentum space thermal
QPs
Cθ(t) =1
σ(t)√
2πexp
[
− θ2
2σ2(t)
]
, σ2(t) = (D/F + a2/2)e2Ft − D/F
Hot QP reaches node after time:
τθ ∼ 1
Fln min
[√
F
D,1
a
]∼
∆30E
2F
|Vqq|2T4
Check
What about next terms in Ck = C0θ + (k − 〈k〉)C1
θ + · · · ? Project out
equation for C1θ:
C1θ (t) = C1
θ (0) exp
[
− tE2F
τθ∆20
]
⇒ assumption of radial equilibrium justified.
Intermediate Temperatures (T ≫ ∆30/E2
F)
|k − kF| ∼√
T∆0
EF≫ ∆2
0
E2F
so outside pair-breaking bottleneck
But
• detailed balance satisfied (both pair-breaking and recombination)
• QPs are created near nodes ⇒ number of antinodal QPs conserved
• Small momentum transfer, q ≪ kF
→ same form of diffusion equation,with different D
τθ ∼E5
F
|V|2√
T 5∆30
with equal contributions from pair-breaking and QP–QP scattering.
Comparison with Experiment
Latest data (Gedik et al. 2004) consistent
with activated behaviour with
Eact ≃ 8 meV
Why does activated behaviour
dominate at such low T?
Compare quantitatively:
(using ∆0EF
≃ 18 and ∆0 ≃ 20 meV for Y123):
Umklapp scattering: τ−1U ∼ |V|2
EFe−Eact/T
Quasielastic scattering: τ−1θ ∼ |V|2
EF
(
∆0
EF
)4(T
∆0
)5/2
(for T ≫ ∆30/E2
F ≃ 3 K)
Only comparable at T ≃ 4 K
⇒ need to measure at lower T to see diffusive process
Conclusions
• Pump-probe experiments:
New information about strongly correlated systems!
How do QP interact? How does SC form dynamically? ...
• Within Fermi liquid picture: bottleneck for pair-breaking.
• Hot QPs relax slowly by scattering from thermal QPs.
• Analytic solution of Boltzmann equation: diffusion in momentum
space and power-law behaviour at low temperature
• Many open questions and challenges from experiments
Role of phonons? (main player in conv. SC)
Main discrepancy: density dependence quantitatively inconsistent
(2 orders of magnitude too weak)
P. C. Howell, A. Rosch, PRL 92, 037003 (2004)